Two different "wave chaotic" systems, involving complex eigenvalues or
resonances, can be analyzed using common semiclassical methods. In particular,
one obtains fractal Weyl upper bounds for the density of resonances/eigenvalues
near the real axis, and a classical dynamical criterion for a spectral gap.Comment: Proceedings of the conference QMath 11; QMath 11, Hradec Kralove :
  Czech Republic (2010

Nonnenmacher, Stéphane

English

arXiv

Proceedings of the conference QMath 11International audienceTwo different ''wave chaotic'' systems, involving complex eigenvalues or resonances, can be analyzed using common semiclassical methods. In particular, one obtains fractal Weyl upper bounds for the density of resonances/eigenvalues near the real axis, and a classical dynamical criterion for a spectral gap

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Quantized open chaotic systemsSte´phane NonnenmacherTo cite this version:Ste´phane Nonnenmacher. Quantized open chaotic systems. Pavel Exner. QMath 11, Sep 2010,Hradec Kralove, Czech Republic. pp.221-226, 2011. <hal-00534698v2>HAL Id: hal-00534698https://hal.archives-ouvertes.fr/hal-00534698v2Submitted on 24 Jan 2012HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.QUANTIZED OPEN CHAOTIC SYSTEMSS. NONNENMACHERAbstract. Two different “wave chaotic” systems, involving complex eigen-values or resonances, can be analyzed using common semiclassical methods.In particular, one obtains fractal Weyl upper bounds for the density of reso-nances/eigenvalues near the real axis, and a classical dynamical criterion for aspectral gap.1. IntroductionThese notes present a sketch of semiclassical methods which can be used todescribe the spectral properties (and as a consequence, the long time properties) ofa certain class of 1-particle quantum chaotic systems. Here are two examples:• damped waves ψ(x, t) on a compact riemannian manifold X of negative sec-tional curvature. The dynamics is described by the damped wave equation(1)(∂2t −∆X + 2b(x)∂t)ψ(x, t) = 0 ,and the damping function b(x) ≥ 0 is assumed to be smooth.• quantum scattering on Rd, described by the Schro¨dinger equation(2) i~∂tψ(x, t) = P (~)ψ(x, t) , P (~) = −~2∆2+ V (x) ,where the potential V (x) has compact support and consists of 3 peakscentered on an equilateral triangle; ~ is Planck’s constant.These two systems seem very different. The configuration spaces are respectivelycompact and of infinite volume, the wave equation does not depend on ~; thedamped wave equation can be rewritten in terms of a contracting semigroup, whilethe propagator e−itP (~)/~ is unitary.In the damped wave situation, Eq. (1) can be diagonalized by a discrete setof metastable modes e−ikj tψj(x), where ψj(x) ∈ L2(X) satisfies the generalizedeigenvalue equation(3)(∆+ k2j + 2i b(x) kj)ψj(x) = 0 .The eigenvalues kj are complex, and lie in the strip −2max(b) ≤ =kj ≤ 0. Weare interested in the distribution of these eigenvalues in the high-frequency limit<kj →∞.In the scattering situation (2), the Hamiltonian P (~) is selfadjoint on L2(R), withabsolutely continuous spectrum on R+. However, the Green’s function (P (~) −z)−1(x, y), well-defined for =z > 0, admits a meromorphic continuation throughR+ to the lower half-plane, with discrete poles {zj(~)} of finite multiplicities (theresonances of P (~)). We will investigate the distribution of these resonances, inthe vicinity of a fixed energy E > 0, in the semiclassical limit ~ → 0. To each12 S. NONNENMACHERresonance is associated a metastable state ψj(x), which is not in L2 but formallydecays with time as e−itzj(~)/~ψj(x).The lifetime of a metastable state is given by the imaginary part of the eigenvalue,τj =12|=kj |, resp. τj =~2|=zj(~)|.Here are some common features of the two systems. In §2.1 we show that thehigh-frequency limit <kj → ∞ is similar with the semiclassical limit ~ → 0. Itis then relevant to study the corresponding classical dynamics: in the dampedwave situation, it is the geodesic flow on X , while in the scattering case it is theHamiltonian flow generated by the Hamiltonian p(x, ξ) = |ξ|22 + V (x), in someenergy interval [E − δ, E + δ]. Our assumptions on X or V (x) imply that theseclassical flows are both “strongly chaotic”.In both cases, the long time properties of our quantum system involves a spec-trum of complex “eigenvalues” associated with metastable states. Our main aim isto understand the distribution of the lifetimes τj in the semiclassical/high energylimit, especially the ones which are not infinitesimally small when ~ → 0: we willthus focus on resonances such that |=zj(~)| = O(~).2. Transformation to nonselfadjoint spectral problemsEach of these two quantum systems can be recast into a spectral problem for anassociated nonselfadjoint differential operator on L2, with discrete spectrum nearthe real axis.2.1. Damped quantum mechanics. Following [1], let us start from the dampedwave system. The generalized eigenvalue equation (3) for <kj  1 can be rewrit-ten using an effective “Planck’s constant” ~ ≈ (<kj)−1, and replacing kj by the“energy” zj =(~kj)22 = 1/2 +O(~):(4) Pdw(~)ψj = zjψj +O(~2), Pdw(~) = −~2∆2− i~b(x) .The principal symbol of the operator Pdw(~), p0(x, ξ) = |ξ|2/2, is real and generatesthe geodesic flow onX . The skew-adjointness of Pdw(~) only appears in the subprin-cipal symbol −i~b(x): the latter does not influence the classical dynamics, but isresponsible for the decay of probability along the flow. Indeed, for ψ0 a wavepacketmicrolocalized on ρ0 = (x0, ξ0) ∈ T ∗X , its evolution ψ(t) = e−itPdw(~)/~ψ0 will beanother wavepacket microlocalized at ρt = (xt, ξt) = Φt(ρ0), with total probabilityreduced by a finite factor(5)‖ψ(t)‖2‖ψ0‖2 ≈ exp(− 2∫ t0b(xs) ds).In the limit ~ → 0, one can speak of a damped classical dynamics: each point ρtevolves according to the geodesic flow, and carries a weight which gets reduced bythe above factor along the flow.The horizontal spectral density of Pdw(~) is given by Weyl’s law, which (to lowestorder) does not depend on the damping [1]: for any c > 0,(6) ]{Spec(Pdw(~)) ∩ ([1/2− c~, 1/2 + c~] + iR)} = ~−d+1(cCX +O(1)),where CX > 0 only depends on X . On the other hand, we will see that the dis-tribution of the imaginary parts =zj(~) strongly depends on the interplay betweenthe geodesic flow and the damping b(x).QUANTIZED OPEN CHAOTIC SYSTEMS 32.2. Complex scaled scattering Hamiltonian and open dynamics. The scat-tering operator P (~) in (2) can be transformed to a nonselfadjoint one through thecomplex scaling method [2]. One deforms the configuration space Rd into a complexcontour Γθ = {x + iθf(x)}, where f(x) = 0 for x in a ball B(0, R) containing thesupport of V (x) (the “interaction region”), while f(x) = x for |x| ≥ 2R. We takean angle θ = M~ log(1/~), M > 0 fixed. This leads to a “scaled” operator Pθ(~),which is no more selfadjoint: in the sector −2θ < arg(z) ≤ 0 it admits discreteeigenvalues, which correspond to the resonances of P (~).Our quest for resonances has turned into the spectral study of Pθ(~). Thisoperator admits the symbolpθ(x, ξ) = p(x, ξ)− iθ〈ξ, df(x)tξ〉+O(θ2)|ξ|2 , p(x, ξ) = |ξ|22+ V (x) .The operator Pθ(~) presents similarities with (4): its principal symbol p(x, ξ) isreal and generates the Hamiltonian flow on p−1(E), while its imaginary part is ofhigher order ~ log(1/~), and generates a damping outside B(0, R). One differencewith Pdw lies in the strength of this damping: for an initial wavepacket localizedon a point ρ0 ∈ p−1(E), the full probability after time t will be reduced by a factor~2M∫t0〈ξs,df(xs)tξs〉ds: the probability is semiclassically strongly suppressed as soonas the trajectory enters the zone where 〈ξ, df(x)tξ〉 > 0 (this “absorbing zone”contains the exterior of B(0, 2R)).Classically, this corresponds to an “open dynamics”: the point ρt evolves ac-cording to the Hamiltonian flow, but it gets “absorbed”, or “killed” as soon as itenters the absorbing zone.Trapped set. For any energy E > 0, the forward (resp. backward) trapped set K−E(resp. K+E ) is defined as the set of initial points which remain bounded for allpositive (resp. negative) times:K∓E = {ρ0 = (x0, ξ0) ∈ p−1(E), |xt| ≤ R, ∀t ≷ 0} ,while the trapped set is made of their intersection KE = K−E ∩K+E . Notice thatKE is a compact flow-invariant set. The above remark shows that any wavepacketlocalized on a point ρ0 6∈ K−E will be absorbed after a finite time, namely the time ittakes to enter the absorbing zone. On the other hand, a point ρ ∈ K−E will convergeto the trapped set KE as t → ∞. This argument shows that, in some sense, longtime quantum mechanics (at energy ≈ E) takes place on KE .3. Fractal Weyl lawsThe above argument can be made precise when estimating the number of reso-nances of P (~) near E. In the case we are interested in, KE is a hyperbolic repeller(that is, there is no fixed point on KE, and all trajectories are hyperbolic), thisnumber is bounded above by a fractal Weyl law directly related with the geometryof KE [4].Theorem 1. Assume that the trapped set KE at some energy E > 0 is a hyperbolicrepeller, and write its Minkowski dimension dimM (KE) = 1 + 2ν. Then for anyc, α > 0, one has in the semiclassical limit(7) ]{Res(P (~)) ∩ ([E − c~, E + c~]− i~[0, α])} = O(~−ν−0) .4 S. NONNENMACHERThis theorem is proved by conjugating Pθ(~) by a suitable “weight”G(~), so thatthe symbol pθ,G of the conjugated operator e−G(~)PθeG(~) satisfies =pθ,G < −2C~outside the√~-neighbourhood of KE . Estimating the volume of this neighbour-hood, and some involved pseudodifferential calculus on pθ,G, lead to the aboveupper bound.A similar argument can be used to study the distribution of decay rates =zj/~for the operator (4). For any time T > 0, one can construct a weight GT (~), suchthat the conjugate operator Pdw,GT (~) = e−GT (~)Pdw(~)eGT (~) admits the symbol(8) pdw,GT (x, ξ) =|ξ|22− i~bT (x, ξ) +O(~2) ,where the subprincipal symbol bT (ρ) = T−1∫ T/2−T/2b(xt) dt is the average of thedamping along the flow. The geometric assumption of negative curvature impliesthat the geodesic flow on X is Anosov, in particular it is ergodic. This implies that,on the energy shell p−1(1/2), the time average bT (ρ) converges almost everywhereto the microcanonical average b¯ = Vol(X)−1∫Xb(x)dx when T → ∞. From thereone can deduce that most of the eigenvalues <zi(~) ∈ [1/2, c~, 1/2+c~] concentratenear the “typical line” =z = −~b¯ [1].Using finer properties of the Anosov geodesic flow, one can estimate the numberof eigenvalues away from this “typical line” [3]. To state the result, we need to intro-duce the extremal ergodic averages of the damping, b− = limT→∞minp−1(1/2) bT ,and similarly for b+.Theorem 2. Assume X is a compact surface of negative curvature. Then, thereexists a function H : R→ R, strictly concave on [b−, b+] and equal to −∞ outside,with maximum H(b¯) = d− 1, such that for any c > 0 and any α ∈ [0, b¯],(9) ]{Spec(Pdw(~)) ∩ ([1/2− c~, 1/2 + c~]− i~[0, α])} = O(~−H(α)−0) .A similar estimate holds for the range −=z ≥ ~α, α > b¯.Comparing this “fractal Weyl upper bound” with the Weyl law (6) confirms thatmost resonances are on the typical line. The above theorem is obtained by studyingthe large deviations of the value distribution of bT in the limit T → ∞: roughlyspeaking, for α < b¯, the volume of the points ρ ∈ p−1(1/2) such that bT (ρ) ≤ αdecays like eT (H(α)−(d−1)). This volume estimate is then used to get (9).In both situations, the upper bounds on counting resonances/eigenvalues wereobtained by deforming the operator Pθ (resp. Pdw) by an appropriate microlocalweight, and studying the imaginary part of the resulting operator by phase spacevolume arguments, where the classical dynamics plays a prominent roˆle.3.1. Spectral gaps. We now present a complementary type of spectral informa-tion, which can be obtained by a similar method for these two systems. Namely,we want to understand if the lifetimes τj(~) can be arbitrarily large in the semi-classical limit; or on the opposite, if there exists a gap of size ∝ ~ between the realaxis and the eigenvalues/resonances. The presence of such a gap has importantconsequences on the long time properties of the system.Let us start with the scattering problem. The gap question can be rephrasedas: “Are the metastable states able to concentrate on KE when ~ → 0?” Theanswer will result from a competition between, one one side, the fast dispersion ofwavepackets due to the hyperbolic classical flow, on the other side the “thickness”QUANTIZED OPEN CHAOTIC SYSTEMS 5of the trapped set allowing the state to reconstruct itself through constructiveinterferences.A dynamical quantity reflecting this competition is of statistical nature, it is atopological pressure associated with the flow on KE:P(−ϕu/2) = limT→∞1Tlog∑γ:T≤Tγ≤T+1exp(−∫ Tγ0ϕu(ρt)/2 dt).Each γ is a periodic orbit in KE with period Tγ , and ϕu(ρ) is the unstable Jacobianof the flow. The above mentioned competition lies in the fact that each exponentialweight gets very small when Tγ → ∞, while the number of terms grows exponen-tially.The following gap criterion was first obtained [5] for the case of hard obstacles,and then generalized in [6] to smooth potentials.Theorem 3. Assume the trapped set KE is a hyperbolic repeller. If the pressureP(−ϕu/2) < 0, then for any c,  > 0 and any small enough ~, the strip[E − c~, E + c~] + i~[P(−ϕu/2) + , 0] does not contain any resonance of P (~).In dimension d = 2, the condition P(−ϕu/2) < 0 is equivalent with a purelygeometric statement, namely the fact that the Hausdorff dimension dimH(KE) < 2(notice that dimHp−1(E) = 3).In the case of damped waves, the gap question is nontrivial if b− = 0, that is, ifthere exists a flow-invariant subset of p−1(1/2) with no damping. A result similarto the one above was obtained in [7]. In this case, the local decay of probability isdue to both hyperbolic dispersion and damping.Theorem 4. Assume that X has negative curvature, and that the topological pres-sure P(−ϕu/2− b) < 0. Then, for any c,  > 0, and ~ > 0 small enough, the strip[1/2−c~, 1/2+c~]+ i~[P(−b−ϕu/2)+, 0] does not contain eigenvalues of Pdw(~).4. Open questionsMost of the above results are upper or lower bounds. The natural question is:“Are these bounds sharp?” The fractal Weyl bound (7) is conjectured to be sharpfor α > 0 large enough, a fact which has been tested numerically on a numberof examples, but could be proved only for a very specific toy model [8]. On theopposite, the bounds (9) for eigenmodes of the damped wave equation are notexpected to be sharp for all values of α. The size of the gap itself is believed tobe larger than the topological pressure bound we gave above. Such an “extra gap”was proved for the 3-disk scattering, using advanced estimates on classical mixing[9].Acknowledgments. The author has been partially supported by the AgenceNationale de la Recherche under the grant ANR-09-JCJC-0099-01.References[1] J. Sjo¨strand, Asymptotic distribution of eigenfrequencies for damped wave equations, Publ.Res. Inst. Math. Sci. 36 (2000) 573-611[2] J. Sjo¨strand and M. Zworski, Complex scaling and the distribution of scattering poles, Journalof AMS 4(1991), 729–769[3] N. Anantharaman, Spectral deviations for the damped wave equation , GAFA 20 (2010) 593-6266 S. NONNENMACHER[4] J. Sjo¨strand and M. Zworski, Fractal upper bounds on the density of semiclassical resonances,Duke Math. J. 137 (2007) 381–459[5] M. Ikawa, Decay of solutions of the wave equation in the exterior of several convex bodies,Ann. Inst. Fourier, 38(1988), 113-14[6] S. Nonnenmacher and M. Zworski, Quantum decay rates in chaotic scattering, Acta Math.203 (2009) 149–233[7] Emmanuel Schenck, Energy decay for the damped wave equation under a pressure condition ,Commun. Math. Phys. 300 (2010) 375–410[8] S. Nonnenmacher and M. Zworski, Distribution of resonances for open quantum maps, Comm.Math. Phys. 269(2007), 311–365[9] V. Petkov and L. Stoyanov, Analytic continuation of the resolvent of the Laplacian and thedynamical zeta function, to appear in Analysis & PDE.Institut de Physique The´orique, CEA/DSM/PhT, Unite´ de recherche associe´e auCNRS, CEA-Saclay, 91191 Gif-sur-Yvette, FranceE-mail address: snonnenmacher@cea.fr

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